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\title{多元统计分析练习4.3-4.4}
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\date{2024 年 4 月 23 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %Problem 01
设两个独立样本 $x_1,x_2,\cdots,x_{n_1}$ 和 $y_1,y_2,\cdots,y_{n_2}$ 分别来自多元正态总体 $N_p(\mu_1,\Sigma)$ 和 $N_p(\mu_2,\Sigma)$. 考虑假设检验 $$H_0:\mu_1=\mu_2, \,\,v.s.\,\, H_1:\mu_1\neq \mu_2. $$
\begin{enumerate}
\item  写出检验统计量和拒绝规则。
\item  写出一切线性组合 $\{a'(\mu_1-\mu_2),a\in\mathbb{R}^p\}$ 的置信度为 $1-\alpha$ 的联合置信区间。
\item  写出线性组合 $\{a_i'(\mu_1-\mu_2),i=1,2,\cdots,k\}$ 的置信度为 $1-\alpha$ 的 Bonferroni 联合置信区间。
\end{enumerate}

\vspace{0.2cm}

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\item  %Problem 02
使用R语言计算例子4.3.1. 表4.2.1和表4.3.1给出了某地区农村的6名2周岁男婴和9名2周岁女婴的身高、胸围和上半臂围。
设总体是多元正态分布，设两总体的协方差矩阵相等。检验2周岁的男婴和女婴的均值向量是否有显著差异。

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\item  %Problem 03
设 $(x_i,y_i), i=1,2,\cdots,n$ 是成对试验的数据，令 $d_i=x_i-y_i$, 又设 $d_1,d_2,\cdots,d_n$ 独立同分布于 $N_p(\delta,\Sigma)$, 其中 $\Sigma>0$, $\delta=\mu_1-\mu_2$ 是总体 $x$ 和 $y$ 的均值向量的差。
考虑假设检验 $$H_0: \delta=0, \,\,v.s.\,\, H_1:\delta\neq 0. $$
写出检验统计量和拒绝规则。 

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\item  %Problem 04
设对同一个单元进行 $p$ 种处理，或在相继的 $p$ 个时间段内重复测量，依次得到测量值 $x_1,x_2,\cdots,x_p$, 其相应的均值依次为 $\mu_1,\mu_2,\cdots,\mu_p$. 将点 $(1,\mu_1),(2,\mu_2),\cdots,(p,\mu_p)$ 用直线连接起来的折线图，成为总体的轮廓。 考虑假设检验，
$$ H_0: \mu_1=\mu_2=\cdots=\mu_p \,\, v.s. \,\, H_1: \mu_i\neq\mu_j, \,\,\mathrm{至少存在一对}\,\, i\neq j. $$
写出检验统计量和拒绝规则。 


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\item  %Problem 05
例子4.4.1. 当施加的处理数 $p=2$ 时，单总体的轮廓分析就退化为基于成对数据的总体均值的比较检验。
用Dalgaard 的 ISWR 中的例子加以说明。（使用关键词 pair 查找成对数据的例子。）
%Figure7.2 具体例子 heart.rate 来说明。

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\item  %Problem 06
设对两个总体的单元分别施加相同的 $p$ 种处理，总体1和总体2的 $p$ 种处理的均值向量分别为 
$\mu_1=(\mu_{11},\mu_{12}, \cdots, \mu_{1p})'$ 和 $\mu_2=(\mu_{21},\mu_{22}, \cdots, \mu_{2p})'$. 
\begin{enumerate}
\item  检验两个轮廓的折线是否平行，即 $$H_{01}: \mu_{1,k}-\mu_{1,k-1} = \mu_{2,k}-\mu_{2,k-1}, 2\le k\le p. $$
\item  在两总体的轮廓平行时，检验两个轮廓的折线是否重合，
即 $$H_{02}: \mu_{11}+\mu_{12}+ \cdots +\mu_{1p} = \mu_{21}+\mu_{22}+ \cdots +\mu_{2p}. $$
\item  在两总体的轮廓重合时，将两个样本合并为一个新样本，检验轮廓是否水平，即 
$$H_{03}: \mu_1=\mu_2=\cdots=\mu_p. $$

\end{enumerate}

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\item  %Problem 07
使用R语言计算例子4.4.2. 作为爱情与婚姻问题某项研究的一部分，对一个由若干名丈夫和妻子组成的样本进行了问卷调查，请他们回答了四个问题。回答均采用1-5分记分制。30名丈夫和30名妻子的回答见表4.4.1. 
总体1定义为``丈夫对妻子''，总体2定义为``妻子对丈夫''。
\begin{enumerate}
\item  画出两个样本的轮廓图。
\item  检验轮廓的平行性。（结果接受轮廓是平行的零假设。）
\item  检验两个轮廓是否重合。（结果接受两个轮廓是重合的零假设。）
\item  检验共同轮廓是否水平。（结果拒绝共同轮廓是水平的零假设。）
\end{enumerate}


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\end{enumerate}


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\end{document}

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